Honors Algebra 2: Extra Practice on Unit 7



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Simplify Completely:

  1. \( \displaystyle i^{1937} + i^{1938} + i^{1939} = \)
  2. Solution

    \( \displaystyle -1\)

  3. \( \displaystyle 4\sqrt{-\frac{7}{12}} - 3 \sqrt{-\frac{1}{24}} \)
  4. Solution

    \( \displaystyle -\frac{i\sqrt{6}}{4} + \frac{2i\sqrt{21}}{3} \)

  5. \( \displaystyle \left( 3 - i \right) ^{-2} \)
  6. Solution

    \( \displaystyle \frac{2}{25} + \frac{3}{50}i \)

  7. Given \( \displaystyle a = -\frac{5}{6} + \frac{1}{2}i\) and \( \displaystyle b = \frac{2}{3} + \frac{3}{4}i,\) find \(\overline{a} + b\)
  8. Solution

    \( \displaystyle -\frac{1}{6} + \frac{1}{4}i \)

  9. Factor \( \displaystyle x^4 + 5x^2 + 6 \) over the complex numbers
  10. Solution

    \( \displaystyle \left( x + i\sqrt{3} \right) \left( x - i\sqrt{3} \right) \left( x + i\sqrt{2}\right) \left( x - i\sqrt{2}\right) \)

  11. Given \( \displaystyle z = -4 + 3i, \) express \( \displaystyle z^{-1} \) as a complex number
  12. Solution

    \( \displaystyle -\frac{4}{25} - \frac{3}{25}i\)

  13. Solve for \(x\) and \(y:\) \( \displaystyle \frac{1}{2}x - 5y - \left(7y - \frac{3}{2}x\right)i = 32 + 45i \)
  14. Solution

    \( \displaystyle x = \frac{1}{4}, y=-\frac{51}{8} \)

  15. Solve for real values of \(x\) and \(y:\) \( \displaystyle 3y + yi = 2i - xi + x \)
  16. Solution

    \( \displaystyle x = \frac{3}{2}; y = \frac{1}{2} \)

  17. Find the sum and difference of \( \displaystyle 7 + 5i \) and its conjugate.
  18. Solution

    Sum: 14; Difference: \( 10i \)

  19. Express each of the following as a complex number given \( \displaystyle a = 5 - 8i; b = -7 + 3i; c = 2 + 8i; d = -4i \)
    1. \( a + b\)
    2. Solution

      \( \displaystyle -2 - 5i \)

    3. \( a + c \)
    4. Solution

      7

    5. \( a - c \)
    6. Solution

      \( \displaystyle 3 - 16i \)

    7. \( c - d \)
    8. Solution

      \( \displaystyle 2 + 12i \)

    9. \( \overline{c} - d \)
    10. Solution

      \( \displaystyle 2 - 4i \)

    11. \( d - \overline{a}\)
    12. Solution

      \( \displaystyle -5 - 12i \)

  20. Factor completely over the complex numbers: \( \displaystyle a^2 + 4 \)
  21. Solution

    \( \displaystyle (a + 2i)(a - 2i) \)

  22. Factor completely over the complex numbers: \( \displaystyle x^4 + 7x^2 + 12 \)
  23. Solution

    \( \displaystyle \left(x + 2i\right)\left(x - 2i\right)\left(x + i\sqrt{3}\right)\left(x - i\sqrt{3}\right) \)

  24. Express the reciprocal of \( \displaystyle 2 - 5i \) as a complex number.
  25. Solution

    \( \displaystyle \frac{2}{29} + \frac{5}{29}i \)

  26. Express \( \displaystyle \frac{2 + 3i}{7 + 4i} \) as a complex number.
  27. Solution

    \( \displaystyle \frac{2}{5} + \frac{1}{5}i \)

  28. \( \displaystyle (4 - 7i)(3 + i) = \)
  29. Solution

    \( \displaystyle 19 - 17i \)

  30. \( \displaystyle \left(3 - 4i \right)^2 = \)
  31. Solution

    \( \displaystyle -7 - 24i \)

  32. \( \displaystyle \left(-1 + i\sqrt{5}\right)^2 = \)
  33. Solution

    \( \displaystyle -4 - 2i\sqrt{5} \)

  34. \( \displaystyle \frac{3 + i}{4 - i} \)
  35. Solution

    \( \displaystyle \frac{11}{17} + \frac{7}{17}i \)

  36. \( \displaystyle \frac{8i}{1 + 3i} \)
  37. Solution

    \( \displaystyle \frac{12}{5} + \frac{4}{5}i \)

  38. \( \displaystyle \frac{3 - 6i}{-2 - 5i} \)
  39. Solution

    \( \displaystyle \frac{24}{29} + \frac{27}{29} i\)